Problem 43 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(260^{\circ}\)

Step-by-Step Solution

Answer

260° is \( \frac{13\pi}{9} \) radians.

Step 1: Understand the Conversion Formula

To convert an angle from degrees to radians, use the formula: radians = degrees × \( \frac{\pi}{180} \). This formula is derived from the fact that 180 degrees is equivalent to \( \pi \) radians.

Step 2: Apply the Conversion Formula

Now apply the formula to convert 260 degrees into radians: 260 × \( \frac{\pi}{180} = \frac{260\pi}{180} \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{260\pi}{180} \). Start by finding the greatest common divisor of 260 and 180, which is 20. Divide both numerator and denominator by 20: \( \frac{260}{180} = \frac{13}{9} \), hence the radians measure is \( \frac{13\pi}{9} \).

Key Concepts

Radians to Degrees

The process of converting radians to degrees is straightforward with a simple formula. Radians and degrees are two units for measuring angles, and converting between them involves understanding their relationship. Since one full circle is equal to \(360^{\circ}\) or \(2\pi\) radians, to convert radians to degrees, you use the conversion factor \(\frac{180}{\pi}\).
Imagine you have an angle measured in radians and you want to see it in a way that's more familiar, using degrees. You just multiply the radian measure by \(\frac{180}{\pi}\). This gives you your angle in degrees.

• Example: Convert \(\pi/3\) radians to degrees.
• Multiply: \(\pi/3 \times \frac{180}{\pi} = 60^{\circ}\)

This transformation is key because degrees are often used in everyday contexts, while radians are common in more advanced mathematics and physics.

Degrees to Radians

When converting degrees to radians, the task is similar but with the reverse conversion factor. Degrees are more familiar, but radians are crucial for calculus and other fields.
The formula to turn degrees into radians is: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\). This multiplier \(\frac{\pi}{180}\) is derived from knowing \(180^{\circ}\) is exactly \(\pi\) radians.
So if you have a degree measurement and need the radian equivalent:

• Take the degree value and multiply by \(\frac{\pi}{180}\).
• Example: Convert \(260^{\circ}\) to radians. \(260 \times \frac{\pi}{180} = \frac{13\pi}{9}\)

Radians offer simplicity in calculations, especially in trigonometry and mathematical analysis.

Conversion Formula

The core of any angle conversion is the conversion formula. Whether converting from degrees to radians or vice versa, these formulas are fundamental to understanding angle measures.
- **Degrees to Radians**: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)- **Radians to Degrees**: \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\)
Why do these formulas work? It's all due to the equivalence relating circles:

• A circle is \(360^{\circ}\) but also \(2\pi\) radians.
• This defines the relationship \(360^{\circ} = 2\pi\) radians.

Thus, these formulas are simply scale factors for converting between the two systems of measurement.
A strong grasp of these formulas is essential for solving many geometry and trigonometry problems successfully.

Greatest Common Divisor

Simplifying fractions is often a key step in angle conversion, particularly when there's a fraction involved in the process. To simplify, knowing about the greatest common divisor (GCD) plays a crucial role.
The GCD of two numbers is the largest number that divides both completely without leaving any remainder. This can dramatically tidy up your fraction’s appearance.
Here's how we simplify \(\frac{260\pi}{180}\):

• Find GCD of 260 and 180, which is 20.
• Divide both numerator and denominator by 20 to get \(\frac{13\pi}{9}\).

By using the GCD, fractions become easier and cleaner, aiding both clarity and accuracy in mathematical work. Simplified results are often more interpretable, particularly in real-world applications.